On deterministic error analysis in variational data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is considered to identify the initial condition. The equation for the error of the optimal initial-value function through the errors of the input data is derived, based on the Hessian of the misfit functional and the second order adjoint techniques. The fundamental control functions are introduced to be used for error analysis. The sensitivity of the optimal solution to the input data (observation and model errors, background errors) is studied using the singular vectors of the specific response operators in the error equation. The relation between "quality of the model" and "quality of the prediction" via data assimilation is discussed

Sensitivity Analysis and Parameter Estimation for Distributed Hydrological Modeling: Potential of Variational Methods

Variational methods are widely used for the analysis and control of computationally intensive spatially distributed systems. In particular, the adjoint state method enables a very efficient calculation of the derivatives of an objective function (response function to be analysed or cost function to be optimised) with respect to model inputs. In this contribution, it is shown that the potential of variational methods for distributed catchment scale hydrology should be considered. A distributed flash flood model, coupling kinematic wave overland flow and Green Ampt infiltration, is applied to a small catchment of the Thor´e basin and used as a relatively simple (synthetic observations) but didactic application case. It is shown that forward and adjoint sensitivity analysis provide a local but extensive insight on the relation between the assigned model parameters and the simulated hydrological response. Spatially distributed parameter sensitivities can be obtained for a very modest calculation effort (6 times the computing time of a single model run) and the singular value decomposition (SVD) of the Jacobian matrix provides an interesting perspective for the analysis of the rainfall-runoff relation. For the estimation of model parameters, adjoint-based derivatives were found exceedingly efficient in driving a bound-constrained quasi-Newton algorithm. The reference parameter set is retrieved independently from the optimization initial condition when the very common dimension reduction strategy (i.e. scalar multipliers) is adopted. Furthermore, the sensitivity analysis results suggest that most of the variability in this high-dimensional parameter space can be captured with a few orthogonal directions. A parametrization based on the SVD leading singular vectors was found very promising but should be combined with another regularization strategy in order to prevent overfitting.JRC.G.9-Econometrics and applied statistic

Assimilation of Remote Sensing Data for River Flows

Abstract. We address the problem of parameters identification and data assimilation for river flows modeled by the 2D St-Venant equations. In practice, available observations are very sparse especially during flood events (very few measurements of elevation at gauging stations in the main channel). We assume we have in addition either surface trajectories extracted from video images (lagrangian data) or space distributed water levels extracted from one satellite image. Then we identify parameters such as the inflow discharge or the topography and/or the initial state (depending on the configuration and the observations available). Numerical twin experiments demonstrate the efficiency of the present method for toy test cases

A weak-constraint 4DEnsembleVar. Part I: formulation and simple model experiments

4DEnsembleVar is a hybrid data assimilation method which purpose is not only to use ensemble flow-dependent covariance information in a variational setting, but to altogether avoid the computation of tangent linear and adjoint models. This formulation has been explored in the context of perfect models. In this setting, all information from observations has to be brought back to the start of the assimilation window using the space-time covariances of the ensemble. In large models, localisation of these covariances is essential, but the standard time-independent localisation leads to serious problems when advection is strong. This is because observation information is advected out of the localisation area, having no influence on the update. This is part I of a two-part paper in which we develop a weak-constraint formulation in which updates are allowed at observational times. This partially alleviates the time-localisation problem. Furthermore, we provide --for the first time-- a detailed description of strong- and weak-constraint 4DEnVar, including implementation details for the incremental form. The merits of our new weak-constraint formulation are illustrated using the Korteweg-de-Vries equation (propagation of a soliton). The second part of this paper deals with experiments in larger and more complicated models, namely the Lorenz 1996 model and a shallow water equations model with simulated convection

Toward the assimilation of images

Abstract. The equations that govern geophysical fluids (namely atmosphere, ocean and rivers) are well known but their use for prediction requires the knowledge of the initial condition. In many practical cases, this initial condition is poorly known and the use of an imprecise initial guess is not sufficient to perform accurate forecasts because of the high sensitivity of these systems to small perturbations. As every situation is unique, the only additional information that can help to retrieve the initial condition are observations and statistics. The set of methods that combine these sources of heterogeneous information to construct such an initial condition are referred to as data assimilation. More and more images and sequences of images, of increasing resolution, are produced for scientific or technical studies. This is particularly true in the case of geophysical fluids that are permanently observed by remote sensors. However, the structured information contained in images or image sequences is not assimilated as regular observations: images are still (under-)utilized to produce qualitative analysis by experts. This paper deals with the quantitative assimilation of information provided in an image form into a numerical model of a dynamical system. We describe several possibilities for such assimilation and identify associated difficulties. Results from our ongoing research are used to illustrate the methods. The assimilation of image is a very general framework that can be transposed in several scientific domains

Estimation de vitesses par assimilation de données variationnelle

Les méthodes classiques d'estimation dense de la vitesse (de type flot optique) s'appuient sur l'estimation des dérivées spatio-temporelles de l'image. Celles-ci sont difficiles à estimer dans le cas d'occlusion d'une partie des acquisitions. Les approches issues de l'assimilation de données s'appuient sur un modèle d'évolution temporelle, qui permet de répondre à ce problème des données manquantes. Nous proposons donc une nouvelle approche pour estimer un champ de vitesse apparent, à partir d'une séquence d'images, en utilisant une méthode d'assimilation de données variationnelle. Pour cela un Modèle Image est construit, dans lequel sont assimilées les observations de la séquence d'images. Cette approche permet une estimation optimale de la vitesse, même si les observations sont partiellement manquantes comme c'est fréquemment le cas en imagerie satellite

Variational assimilation of Lagrangian data in oceanography

We consider the assimilation of Lagrangian data into a primitive equations circulation model of the ocean at basin scale. The Lagrangian data are positions of floats drifting at fixed depth. We aim at reconstructing the four-dimensional space-time circulation of the ocean. This problem is solved using the four-dimensional variational technique and the adjoint method. In this problem the control vector is chosen as being the initial state of the dynamical system. The observed variables, namely the positions of the floats, are expressed as a function of the control vector via a nonlinear observation operator. This method has been implemented and has the ability to reconstruct the main patterns of the oceanic circulation. Moreover it is very robust with respect to increase of time-sampling period of observations. We have run many twin experiments in order to analyze the sensitivity of our method to the number of floats, the time-sampling period and the vertical drift level. We compare also the performances of the Lagrangian method to that of the classical Eulerian one. Finally we study the impact of errors on observations.Comment: 31 page

A reduced-order strategy for 4D-Var data assimilation

This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EO F analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a mul tivariate background error covariance matrix $\textbf{B}_r$, and an important decrease of the computational burden o f the method, due to the drastic reduction of the dimension of the control space. % An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of $\textbf{B}_r$ brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method

An adjoint method for the assimilation of statistical characteristics into eddy-resolving ocean models

The study investigates perspectives of the parameter estimation problem with the adjoint method in eddy-resolving models. Sensitivity to initial conditions resulting from the chaotic nature of this type of model limits the direct application of the adjoint method by predictability. Prolonging the period of assimilation is accompanied by the appearance of an increasing number of secondary minima of the cost function that prevents the convergence of this method. In the framework of the Lorenz model it is shown that averaged quantities are suitable for describing invariant properties, and that secondary minima are for this type of data transformed into stochastic deviations. An adjoint method suitable for the assimilation of statistical characteristics of data and applicable on time scales beyond the predictability limit is presented. The approach assumes a greater predictability for averaged quantities. The adjoint to a prognostic model for statistical moments is employed for calculating cost function gradients that ignore the fine structure resulting from secondary minima. Coarse resolution versions of eddy-resolving models are used for this purpose. Identical twin experiments are performed with a quasigeostrophic model to evaluate the performance and limitations of this approach in improving models by estimating parameters. The wind stress curl is estimated from a simulated mean stream function. A very simple parameterization scheme for the assimilation of second-order moments is shown to permit the estimation of gradients that perform efficiently in minimizing cost functions